The DUNE-DPG library for solving PDEs with Discontinuous Petrov--Galerkin finite elements

TL;DR

The paper introduces Dune-DPG, a flexible C++ library built on Dune for solving PDEs using the Discontinuous Petrov--Galerkin method, ensuring stability and error control for a wide range of problems.

Contribution

It presents a modular, versatile C++ library that implements DPG formulations for various PDEs, extending beyond classical problems.

Findings

Successfully solves diverse PDE classes including transport problems

Provides stable variational formulations with reliable error bounds

Enables applications like optimal control using DPG

Abstract

In the numerical solution of partial differential equations (PDEs), a central question is the one of building variational formulations that are inf-sup stable not only at the infinite-dimensional level, but also at the finite-dimensional one. This guarantees that residuals can be used to tightly bound errors from below and above and is crucial for a posteriori error control and the development of adaptive strategies. In this framework, the so-called Discontinuous Petrov--Galerkin (DPG) concept can be viewed as a systematic strategy of contriving variational formulations which possess these desirable stability properties, see e. g. Broersen et al. [2015]. In this paper, we present a C++ library, Dune-DPG, which serves to implement and solve such variational formulations. The library is built upon the multipurpose finite element package Dune (see Blatt et al. [2016]). One of the main features of Dune-DPG is its flexibility which is achieved by a highly modular structure. The library can solve in practice some important classes of PDEs (whose range goes beyond classical second order elliptic problems and includes e. g. transport dominated problems). As a result, Dune-DPG can also be used to address other problems like optimal control with the DPG approach.